Abstract:We study exact exponential-time algorithms for Edge Deletion to Cactus. Given a connected graph $G$, the task is to delete a minimum number of edges so that the remaining spanning graph is a connected cactus. Akhtar and Philip (IWOCA 2026) gave an $O^*(3^n)$-time algorithm for the unweighted problem, where $n$ is the number of vertices in the input graph and the $O^*(\cdot)$ notation hides polynomial factors.
We improve this bound to $O^*(2^n)$ time and space. More generally, if the deletion costs take at most $q$ distinct nonnegative real values, then the weighted problem can be solved in $O^*(2^n n^{O(q)})$ time and space. Thus every fixed number of distinct costs, and in particular the unweighted case, admits a faster exact algorithm. For nonnegative integer costs of total weight $W$, we obtain an $O^*(2^n(W+1))$ pseudo-polynomial algorithm, while arbitrary nonnegative real costs admit an $O^*(3^n)$ exact algorithm.
From: Wenhao Song [view email]
[v1]
Tue, 16 Jun 2026 06:27:51 UTC (13 KB)
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